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Countable contraction mappings in metric spaces: invariant sets and measure

100%
Barrozo M., Molter U.
Open Mathematics
|
2014
|
tom 12
|
nr 4
593-602
EN
We consider a complete metric space (X, d) and a countable number of contraction mappings on X, F = {F i: i ∈ ℕ}. We show the existence of a smallest invariant set (with respect to inclusion) for F. If the maps F i are of the form F i(x) = r i x + b i on X = ℝd, we prove a converse of the classic result on contraction mappings, more precisely, there exists a unique bounded invariant set if and only if r = supi r i is strictly smaller than 1. Further, if ρ = {ρ k}k∈ℕ is a probability sequence, we show that if there exists an invariant measure for the system (F, ρ), then its support must be precisely this smallest invariant set. If in addition there exists any bounded invariant set, this invariant measure is unique, even though there may be more than one invariant set.
2
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Minimization of the number of periodic points for smooth self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers

100%
Graff G., Kaczkowska A.
Open Mathematics
|
2012
|
tom 10
|
nr 6
2160-2172
EN
Let f be a smooth self-map of m-dimensional, m ≥ 4, smooth closed connected and simply-connected manifold, r a fixed natural number. For the class of maps with periodic sequence of Lefschetz numbers of iterations the authors introduced in [Graff G., Kaczkowska A., Reducing the number of periodic points in smooth homotopy class of self-maps of simply-connected manifolds with periodic sequence of Lefschetz numbers, Ann. Polon. Math. (in press)] the topological invariant J[f] which is equal to the minimal number of periodic points with the periods less or equal to r in the smooth homotopy class of f. In this paper the invariant J[f] is computed for self-maps of 4-manifold M with dimH 2(M; ℚ) ≤ 4 and estimated for other types of manifolds. We also use J[f] to compare minimization of the number of periodic points in smooth and in continuous categories.
3
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A cohomological index of Fuller type for parameterized set-valued maps in normed spaces

68%
Skiba R.
Open Mathematics
|
2014
|
tom 12
|
nr 8
1164-1197
EN
We construct a cohomological index of the Fuller type for set-valued flows in normed linear spaces satisfying the properties of existence, excision, additivity, homotopy and topological invariance. In particular, the constructed index detects periodic orbits and stationary points of set-valued dynamical systems, i.e., those generated by differential inclusions. The basic methods to calculate the index are also presented.
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